Example 3.1.3.
Suppose we roll a fair 6-sided die two times, and let \(S\) be the sum of the rolls. It’s straightforward to check the distribution for \(S\) shown below. We’ve also included a column containing the products \(k \Pr(S = k)\text{,}\) which must be summed up to find \(\E(S)\text{.}\)
| \(k\) | \(\Pr(S = k)\) | \(k \cdot \Pr(S = k)\) |
|---|---|---|
| 2 | \(1/36\) | \(2/36\) |
| 3 | \(2/36\) | \(6/36\) |
| 4 | \(3/36\) | \(12/36\) |
| 5 | \(4/36\) | \(20/36\) |
| 6 | \(5/36\) | \(30/36\) |
| 7 | \(6/36\) | \(42/36\) |
| 8 | \(5/36\) | \(40/36\) |
| 9 | \(4/36\) | \(36/36\) |
| 10 | \(3/36\) | \(30/36\) |
| 11 | \(2/36\) | \(22/36\) |
| 12 | \(1/36\) | \(12/36\) |
The sum of the third column gives \(\E(S) = \frac{252}{36} = 7\text{.}\)